Fourth-order tensors

Abstract

The focus of this report is to study fourth-order tensors and generalize some results from the linear algebra of second-order tensors to fourth-order tensors. A fourth-order tensor can be viewed as a linear map from second-order tensors to second-order tensors. We provide an orthonormal basis for the vector space of fourth-order tensors and use it to represent any fourth-order tensor by a fourth-dimensional array, which represents its components' form. An inner product and norm are provided for this vector space. Composition of linear maps gives rise to multiplication of fourth-order tensors, which we present in components' form. We study different kinds of symmetries for fourth-order tensors, in particular, major symmetry and minor symmetry. We provide an isomorphism between the vector space of fourth-order tensors and the vector space of second-order tensors of the same dimension. We use this isomorphism to prove a spectral theorem for fourth-order tensors that possess major symmetry.